Study of the QGP physics in center vortices Electric and magnetic after/before center vortex removal T. Saito (Kochi Univ.) in collaboration with Y. Nakagawa (Niigata Univ.) M.N. Chernodub (Univ. of Tours, Univ. of Gent, ITEP) A. Nakamura(Hiroshima Univ.) V.I. Zakharov(ITEP, Max Planck Institute) Lattice 2010 June 14-19 Villasimius, Sardinia, Italy The XXVIII International Symposium on Lattice Theory

1 gluon plasma

 Heavy-ion experiment (RHIC,2004) produces a quark gluon plasma in the deconfinement phase.  Not free gas, but strongly interacting matter.  Perfect fluid (Hydro calc. and elliptic flow)  Small shear viscosity (also by lattice simulations)  Strongly interacting QGP; however, understanding of dynamics of QGP is an unsoved problem, so that we need new idea.  Here, we study what magnetic degrees of freedom causes in the QGP phase.

2 Magnetic degrees of freedom in hot Yang-Mills theory (1) Magnetic in hot medium produces magnetic mass (~g^2(T)T) as an infrared cutoff of thermal QCD. Linde PLB96,289(1981); for lat. cal. , Nakamura, et. al, PRD69 (2004) 014506 (SU(3)) and Heller, et. al; PRD57,1438,(1998) (SU(2)). Spatial- above the critical temperature shows a linearly rising potential and its thermal string tension scales as magnetic couplings. Bali, et. al, PRL71,3059(1993) Gribov-Zwanziger (GZ) confinement scenario also shows the remnant of confining in QGP and its non-vanishing string tension depends on magnetic scaling. Nakagawa, PRD73 (2006) 094504 Gribov-type dispersion relation may work well in the deconfinement phase. Zwanziger,PRL94(2005)182301;PRD76(2007)125014

3 Magnetic degrees of freedom in hot Yang-Mills theory (2) Center vortices (as a of SU(N) ) are responsible for non-perturbative phenomena such as and chiral breaking. This idea has originally been discussed by t’Hooft, Mack, Cornwall, etc. Role of magnetic monopole in hot medium has been studied by Chernodub and Zakharov and they explain the dynamics of sQGP. PRL98(2007)082002 ; also, Shuryak gives a similar idea based on monopole. PRC75(2007)054907; PPNP62(2009)48. Center vortices in the hot medium; Chernodub, Nakamura, Zakharov, PRD78,074021(2008) However, there is no clear connection between thermal gluons as a basic element of QCD (or QGP) and topological objects.

4 Center vortices on the lattice

 Center vortices are responsible for the color confinement and chiral symmetry breaking.  Center vortices as topological defect are defined via center group Z(N) of the gauge group SU(N).  Lattice numerical simulations support Identify center vortices on the lattice Vortex density shows asymptotic scaling Phase transition and spatial string tension

Illustration of vortex- monopole chain; Chernodub, et. Al, PRD78:074021,2008

5 Maximal center projection  Numerical technique Direct Maximal Center Projection (MCP) by Debbio, et. al, PRDv58,094501  We apply the MCP to all configurations of the SU(2) gauge field

1 2 All the Us  I Maximize R  TrU  (x,t) VT x,t

Z (x)  sgn TrU (x)

 Removing center vortex (via de Forcrand – D’Elia procedure,PRL82,4582(1999))

U (x) U (x)  Z (x)U (x)

 Vortices carry the non-perturbative IR physics of QCD  Handling vortices numerically enables us to switch on/off non-perturbative mode !! In particular, this technique shall be applied to the QGP physics.

6 Gluon propagators at finite temperature Self-energy and propagators 1 1  k  k   GP  FP  D  P  P    0: Landau gauge T L G  k 2 T F  k 2 L k 2 k 2 Projection operators k ik j k  k P00  P0i  Pi0 , Pij   ij   , P      P, T T T T k 2 L k 2 T Electric and magnetic gluon propagators in the momentum space

2 2 (PT )  PT , (PL )  PL , PT PL  0

 1  1 D (k,k  0)  D00   , D (k,k  0)  Dii   E 0 F  k 2 M 0 G  k 2   2 F(0,0)  mE ~ g(T)T G(0,0)  mM ~ g (T)T

7 Gluon propagators on the lattices

Gauge potentials and correlators

a a * A (x,t)  Tr U (x,t) D(x,t)  A (x,t)A (x,t)

Unequal-time propagators ( t  t '  t ' ' )

 1 a ' a '' iq(x y) D(q,t)  2  A (x,t )A (y,t )e V (Nc 1) x

Sum of t with q0=0,  1  D(q,q0  0)   D(q,t) Nt t

But note that for the Coulomb gauge equal-time gluon ( t '  t '') will be calculated because of unfixing the temporal direction of gauge field.

8 Numerical setting

 SU(2) lattice calculation with quenched Wilson-gauge action  Landau (Coulomb) gauge on the lattice in the path-integral formula satisfies the following condition: 2 1 a eps Tr U (x) U (x  ˆ) 10  A (x,t)  0  Maximize R  ReTrU (x,t)      VT x,t  Wilson-Mandula Method (PLB185,127(1987)) Parameters:  Lattice size : 24x24x24x4  beta : 2.2-2.6, corresponding to the temperatures T/Tc are approx. 1.40, 3.00 and 6.00.  Configurations: 10k dicarded and 20-30 confs. circa are used to measure.  Convergence criteria: eps = 10^(-08) for gauge fixing and eps = 10^(- 16) for maximal center projection. Procedure: Gauge updated --> Maximal center projection --> Gauge fixing

9 Gluon propagators in the Landau gauge(1)

 Magnetic gluons drastically change in infrared regions  At higher T (LHC temp.), the center vortices affect the gluon propagators.

10 Gluon propagators in the Coulomb gauge(1)

T /Tc 1.40

Gribov-Zwanziger confinement scenario for the Coulomb gauge QCD survives.

Time-time (electric) correlator diverges in the infrared limit.  Instantaneous linearly rising potential and non-zero thermal string tension that depends on magnetic scaling. Spatial-Spatial (magnetic) correlator is suppressed in the infrared limit. The same behavior occurs in the deconfinement phase; there exists

confinement caused by magnetic degrees of freedom in the QGP phase. 11 Summary of this talk

 We have studied the sQGP physics from the hot gluon propagators via the lens of center vortex mechanism.  We found that magnetic degrees of freedom are so important via the study of the gluon propagators after/before center vortices; conversely center vortices are still important objects above Tc.  In Landau and Coulomb gauges, we get similar results; that is, the magnetic sector is very singular even in the QGP phase.  Magnetic degrees of freedom cause (magnetic) confinement in the deconfinement phase. We have to consider this point properly to understand sQGP.

12 Future plans

 More temperature dependence and volume dependence.  Calculation of trasport coefficient after/before center removal; it reveals the reason of small shear viscosity. ( in progress)  Equation of state after/before center removal; how strongly the non-perturbative mode influences on the EOS.

13 Backup slides

14 Effect for a numerical ambiguity of the maximal center (gauge) procedure

after some random gauge transformations

 The procedure “direct maximal center gauge (projection)” (Debbio, et. al, PRDv58,094501 ) produces a lot of gauge ambiguities (copies); but the influence to the gluon propagators is very small.

基研研究会「熱場の量子論とその応用」 15 2009 Spatial Wilson loop in the QGP

 Spatial Wilson loop gives a linearly rising potential in the QGP.

G.S. Bali, et. al, PRL71,3059(1993)

 (T)  cg 2 (T)T W(R,S) ~ exp( s RS ) s

T  Tc

16 Linearly rising potentials Removing vortices eliminates confinement.

This figure is from “SU(2) gluon propagators from the lattice – a preview”, hep- lat/0104003, Kurt Langfeld

The result changes drastically after vortices removal

Modified configurations have only perturbative Confinement region (T=0) propertis

All the non-perturbative infrared physics must be carried by {Z}.

17 Example for center vortex removal

 Numerical study of Coulomb gauge QCD via center vortex (Greensite, Olejnik, Zwanziger, PRD69, 074506 (2004) )

Gribov-Zwanziger scenario in the Coulomb gauge QCD: Instantaneous interaction (link-link correlator on the lattice ) produces a confining potential even in the QGP phase.

T /Tc ~1.40

Large difference, Polyakov line correlators because this is magnetic.

Small difference,

 because this is V(R,0)  log(U(R,0)U (0,0)) electric.

18